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In a Dutch oven, heat olive oil over medium heat.
Add onion, carrots, ½ teaspoon salt, and pepper to a large pot or Dutch oven. Cook, stirring occasionally, until softened, about 5 minutes.
Add garlic; cook, stirring often, 2 minutes.
Stir in flour; cook, stirring constantly, 2 minutes.
Gradually whisk in stock, breaking up any lumps. Stir in half and half.
Increase heat to medium-high to bring to a simmer. Reduce heat to low.
Add broccoli and remaining ½ teaspoon salt to pan. Cover, and cook at a low simmer until broccoli is tender, about 15 minutes.
Stir in shredded cheddar and mustard until smooth. Taste, and add salt, if desired. Garnish servings with cheddar.